Two main themes describe my current research interests:
- Ergodic theory of dynamical systems and its applications to the analysis of biological processes at molecular, cellular and population levels.
- Quantum statistics as a formalism to investigate the dynamics of electron transport and proton transduction in cellular metabolism.
These themes involve work of a methodological nature — which is primarily mathematical — and studies of biological models using computational and empirical methods. The methodological studies draw from areas such as the theory of large deviation, non-linear dynamical systems, game theory, products of random matrices. The biological processes investigated include: evolutionary genetics and evolutionary dynamics, biological networks, allometric relations and demography.
Methodological studies: An important challenge in the study of biological systems is to understand in quantitative terms the relation between macroscopic patterns and processes, and the behavior of the individual components of the system. This problem arises, for example, in studies of life-history evolution, where the issue is to understand the evolutionary dynamics of life-history variables in terms of the mechanisms which operate primarily on the individual birth and death rates. The analysis of this problem has led to the development of a mathematical structure — called the evolutionary formalism — which studies the relation between microscopic variables and related macroscopic parameters in certain classes of dynamical processes in biology.
One of the tenets of the evolutionary formalism is the following analytical fact: The growth rate parameter in models of certain classes of dynamical systems in biology satisfies a variational principle which is formally analogous to the minimization of the free energy in statistical thermodynamics. This principle implies a precise correspondence between certain concepts in thermodynamic theory, and certain macroscopic variables that characterize the behavior of dynamical systems in biology. The evolutionary formalism has led to the discovery of the concept evolutionary entropy, an analogue of the Gibbs-Boltzmann entropy, as a descriptor of the structure and behavior of certain classes of biological systems at the molecular, cellular and population levels. The variational principle entails that the methods of equilibrium statistical mechanics, which revolve around thermodynamic concepts such as free energy and temperature, can be exploited to study the non-equilibrium behavior of biological systems, which are described by processes defined by parameters such as growth rate and cycle time.
Biological networks: Large scale studies in functional genomics now show that the networks which describe the gene-regulatory systems, protein-DNA interactions, signal transduction pathways are characterized by certain statistical signatures, in particular robustness, the capacity of the network to remain functional in the face of random deletion of nodes and edges. Elucidating the relation between the topology of the networks and their statistical properties has emerged as one of the central problems in the new activity called Systems Biology. These problems are currently being addressed in terms of the evolutionary formalism. The important parameter which has emerged from these studies is the concept, network entropy — a special case of the evolutionary entropy concept. Analytical and computational studies have shown that network entropy, a quantitative measure of the rate of information flow within the network is a precise measure of the property robustness. This relation between entropy and robustness is being used to explore the relation between the structure and function of biological networks.
Life-history evolution: One of the central issues in evolutionary genetics is the development of quantitative models to explain the diversity of life-history in natural population, that is, the large variability in fecundity and mortality rates which exist within and between species. R.A. Fisher, one of the pioneers in evolutionary theory, realized that any solution of this problem requires a quantitative measure to predict the outcome of competition between an invading type and the resident population, and proposed the population growth rate — the Malthusian parameter — as the predictor of competitive success. Since Fisher’s proposal the population growth rate has become the dominant parameter in both theoretical and empirical studies of life-history evolution.
Studies based on the ergodic theory of dynamical systems and diffusion processes showed that growth rate determines invasion success only in populations of effectively infinite size: In finite populations it was shown that the dynamics of invasion is a stochastic process which is predicted by the parameter evolutionary entropy, a measure of the demographic stability or robustness of the population. This demographic parameter has been integrated with Mendelian genetics to develop a dynamical theory of evolution called directionality theory. This theory predicts relations between ecological constraints and life-history variables and provides a framework for explaining the diversity of physiological and morphological properties which exist in natural populations.
Allometric relations: Allometric studies which began with the work of Kleiber in 1930’s show that the metabolic rate of organisms: uni-cells, plants and animals satisfies certain scaling laws with respect to body mass. I have developed a class of models to explain these empirical rules. These models, in sharp contrast to earlier attempts to addresss this problem, explain both the diversity in proportionality parameters and the variation in scaling exponents observed. The models recognize that energy transduction in organisms occurs by means of electron transfer between redox centers in biomembranes and that this electron transfer process occurs by quantum tunnelling. The methods of quantum statistics were exploited to derive a scaling relation between the metabolic flux in biomembranes and the cycle time of the metabolic process in uni-cells. The scaling relation is the basis for investigating both theoretically and empirically, relations between body size, and physiological and life-history variables such as metabolic rate and life span, respectively.
The origin and evolution of aging: Maximal life span potential is defined as the maximum observed life span of a species. There is about a 50-fold range of variation of this parameter within the mammalian species. The problem of explaining this range of variation has been an important issue in studies of gerontology. Efforts to address this problem and to explain the large differencies in the rate of aging between species have been driven to a large extent by the rate of living theory oxidative stress theory. This theory essentially asserts that metabolic rate determines the rate of aging. I have appealed to new models of the aging process to propose the hypothesis that metabolic stability, the capacity of metabolic networks in the cell to maintain steady state concentrations of metabolites, is the prime determinant of aging. This new class of models has been integrated with evolutionary models to develop a new theory of aging which is able to (a) predict the maximum life span potential of species, (b) evaluate the effect of interventions such as caloric restriction on species life span.